Due to Apple's security and app store policies, converting YouTube files into MP3 on iPhone requires some extra steps compared to other devices. While it is feasible, your process is severely limited. Nearly all YouTube to MP3 converters are web apps that operate perfectly on computers, but not at any applicable or fast speed as iOS human nature has limitations.
For example, YTMP3 brings free video conversion to the web and can do it directly in Safari with a nice search that integrates itself into websites like Soundcloud. If you download a file from the internet, it's not as simple to tell your iOS iPhone (or even iPad) where that data should go so we'll need some sort of app or service like iCloud Drive. To get around this, iPhone users typically a required to use an ancillary application that can usually download and manage files (I suggest Documents by Readdle). Users download the mp3 file to then transfer it in the Music app or another player.
This method works with varying degrees of efficiency. If you use one of the built-in tools, conversions will happen at about 3-5 seconds per minute of video but there's some added file system limitations in iPhone so it is slightly longer. There is a 20-percent higher error rate for those thingt to download media files directly via Safari according from the report of users on this converters because it enforces pop-ups and redirects.
The type of tools are also largely dependent on the Apple ecosystem. The apps cannot break YouTube's terms of service, and therefore Apple will not approve them for the App Store. This failing of app support also makes browser-based solutions or third-party file managers a must-have for iPhone users. Apple CEO Tim Cook also stressed Apple's commitment to user privacy and security, which is one reason why these are procedural bugs that for many apps turn out as well.
The question of legality also remains a paramount issue. Unlicensed downloads continue to be a concern for the music industry. Converting copyrighted content to a MP3 file without permission from the copyright owner may violate YouTube's Terms of Service and could result in legal action against you. Apple provides the infrastructure to give this an enabler by not allowing apps that can help it, thereby caging many users in legal boundaries.
Quick-fix desktop tools are fine, but there's more action in that market on the web these days and unfortunately iOS users don't have a better option than YouTube to MP3 : it will just involve a little bit of extra legwork—and possibly some limitations as well.